Of Central Forces. 419 
the apfides, For making the equation of limits =o, we 
have be al P x if ?—3_ , Whence it is evi- 
M—t1 u—t1 
dent, that there can be no more than four roots, two 
pofitive and two negative. But to difcover in every cafe 
the number of poffible roots is a problem of confiderable 
difficulty, Dr. Waring has pointed out the method of 
doing this (Meditationes Algebraica Prob, 14,); but as his 
manner of writing is in general very concife, an eafy 
inveftigation of the feveral conclufions there deduced may 
not be improper in this place, 
_ It is well known, that if by varying the coefficients of 
an equation two roots become equal, the next inftant they 
will be impoffible, and immediately before becoming Sauk} 
they will be real and unequal. 
This being granted, let it be Breer to find the number 
of poffible roots in the equation x” — Ax™ — B=», which 
isan-equation of the fame kind with that for determining 
the apfides, Firft, find the equation of limits, and make ig 
1 
a m | 2—-m 
= 0, viz, nx”? — m Ax" = 0, then x = — 
n 
A —., hence x has two values, if 2 and m be both even 
numbers, and — A a negative number; wherefore the 
number of real roots in the given equation cannot exceed 
four, 
Multiply the firft equation by w, and ihe fecond by x, 
and take the difference of the produéts, then m—n x Axm 
1 
ou Me: 
aa heey y which 
will give two other limits, if — B be pofitive, and the 
reft as above, 
If the given equation have two equal roots, they will 
€oincide both with the firft and fecond values of « jut 
Fifa found 
nN 
—nB=o0; from whence x — 
